Spherical Foams in Flat Space

TL;DR

This paper explores the geometry and stability of foam structures derived from regular tessellations on a 3-sphere, using conformal projections to understand their shape, distribution, and equilibrium properties in flat space.

Contribution

It introduces a novel approach to model irregular foam cells via conformal projection from S^3, extending understanding of foam structure and stability beyond Euclidean tessellations.

Findings

Characterized classes of foam bubbles on S^3

Linked foam shape distribution to conformal geometry

Identified a local maximum in foam stability at tetrahedral bubbles

Abstract

Regular tesselations of space are characterized through their Schlafli symbols {p,q,r}, where each cell has regular p-gonal sides, q meeting at each vertex, and r meeting on each edge. Regular tesselations with symbols {p,3,3} all satisfy Plateau's laws for equilibrium foams. For general p, however, these regular tesselations do not embed in Euclidean space, but require a uniform background curvature. We study a class of regular foams on S^3 which, through conformal, stereographic projection to R^3 define irregular cells consistent with Plateau's laws. We analytically characterize a broad classes of bulk foam bubbles, and extend and explain recent observations on foam structure and shape distribution. Our approach also allows us to comment on foam stability by identifying a weak local maximum of A^(3/2)/V at the maximally symmetric tetrahedral bubble that participates in T2 rearrangements.